hencesupn∈NEP[|Xn|τ] < +∞follows from (148), (149) and EP|B|τ ≤ C′EQ|B| < ∞(with someC′, this is a consequence ofdQ/dP ∈ W).
4.4 A digression – back to discrete time
Using the arguments for continuous-time markets it is possible to prove a complement to Theorem 3.16: we can replace Assumption 3.12 by (135). During this brief section we get back to the discrete-time setting of Chapter 3.
Theorem 4.16. Let Assumptions 2.19, 3.2 and 3.11 be in force. Let furthermoreB ∈L1(P), V−(z−B)<∞,u−, w−non-decreasing and
α < β, α
γ <1<β δ hold. Then
sup
θ∈A(z)
V(θ, z)<∞. and there existsθ∗∈ A(z)with
sup
θ∈A(z)
V(θ, z) =V(θ∗, z).
Proof. Corollary 2.47 providesQ∈ MwithdQ/dP ∈L∞,dP/dQ ∈ W. Clearly, B ∈ L1(Q).
We claim that for eachθ ∈ A(z) := {θ ∈ Φ : V−(θ, z) <∞} we also haveXTz,θ ∈ L1(Q), i.e.
the strategyθis also inA(z)as defined in (128) of the present chapter (this explains why we did not seek new notation). Indeed,V−([XTz,θ−B]−)<∞and Lemma 4.12 withs:= 1< β/δ imply thatEP[[XTz,θ−B]−]<∞hence alsoEP([XTz,θ]−)<∞. BydQ/dP ∈L∞we also have EQ([XTz,θ]−)<∞. Proposition 5.3.2 of [54] (see also its proof) entails thatXtz,θ,t= 0, . . . , T is aQ-martingale, in particular,XTz,θ ∈L1(Q).
Then the argument for proving Theorem 4.15 implies thatsupn→∞V−([Xn−B]−)<∞for any maximising sequenceXn =XTz,φ(n). Notice thatw−(x)> 0, x > 0 by Assumption 3.11 hence, by the proof of Lemma 3.7, we get that(φ1(n), . . . , φT(n))is a tight sequence. Now we can conclude just like in the proofs of Theorems 3.4 and 3.16 above, using Assumption 3.2.
Remark 4.17. It is interesting to note that the proofs of tightness in Theorems 3.16 and 4.16 follow entirely different ideas. In Theorem 3.16 we manage to find an EUT optimal investment problem whose value function is above that of the CPT problem. In Theorem 4.16 we useQto find estimates forXTz,θwhich then translate into estimates for(φ1(n), . . . , φT(n)).
4.5 Existence
We return to the setting of Theorem 4.15. Letνndenote the joint law of the random vector (ρ, Xn). As a consequence of Theorem 4.15, the sequence {νn;n∈N} is also tight. and we can extract a weakly convergent subsequence{νnk;k∈N}with limit πfor some probability measureπonR2.
We recall that a mappingK fromR× B(R)into[ 0,+∞) is called a transition probability kernel on a probability space(R,B(R), µ)if the mappingx7→K(x, B)is measurable for every setB∈ B(R), and the mappingB7→K(x, B)is a probability measure forµ-a.e.x∈R.
By the disintegration theorem (see e.g. [36]) there exists a probability measureλonRand a transition probability kernelK on(R,B(R), λ)such thatπ(A1×A2) = R
A1K(x, A2)dλ(x) for allA1, A2∈ B(R).Clearly,λ(A) =P(ρ∈A)for all Borel setsA⊆R.
Theorem 4.18. Under Assumptions 3.11, 4.1, 4.2, 4.3, 4.4 andV−(z−B)<∞there exists an optimal trading strategyφ∗=φ∗(z)for(129).
Proof. AsB is replicable by Assumption 4.4 with a replicating portfolio, say,ψ, we may re-placeφn byφn−ψand assumeB= 0.
Let us setX∗ :=G(ρ, U∗), whereGis the measurable function given by Lemma 6.16 ap-plied withδ:=λ,ν :=KandY :=ρ. Clearly, the random variableX∗isσ(ρ, U∗)-measurable.
Moreover, the subsequence of random variables Xnk, k∈N converges in law to X∗ as k→+∞. So
u+ Xn+k k∈Nalso converges in law tou+(X∗+). Hence,limkP u+ Xn+k
> y
= P(u+(X∗+)> y) for everyy ∈ Rat which the cumulative distribution function of u+(X∗+)is continuous (i.e. outside a countable set). Analogously, we conclude thatP u− Xn−k
> y
Secondly, we prove that the inequality V(X∗) ≥ V∗ holds. We already know, from the previous step, thatV−(X∗−) ≤ lim infkV− Xn−k
. We note further that, by the proof of The-orem 4.15, supk∈NEP
fory >1, we see thatgis an integrable function on[ 0,+∞). It follows from Markov’s inequal-ity thatw+ P u+ Xn+k find real-valued random variablesY andYk, withk∈N, on some auxiliary probability space Ω,ˆ Fˆ,Qˆ
, such that each Yk has the same law asρXnk, Y has the same law as ρX∗, and Yk →Y Q-a.s., asˆ n→ ∞. It is then clear thatEQ[Xnk] =EP[ρXnk] =EQˆ[Yk]for everyk∈N.
We know from the proof of Theorem 4.15 thatsupk∈NEP
h
Xn−kξi
<+∞, for some1< ξ < βδ. Consequently, we can chooseϑ >1such thatϑ < ξ, and using H¨older’s inequality we obtain
EQˆ Vall´ee-Poussin criterion, the familyYk−,k∈Nis uniformly integrable and thus
k∈NlimEQˆ
Yk−
=EQˆ
Y−
<+∞. (151)
Furthermore, using Fatou’s lemma we get the inequalityEQˆ[Y+]≤lim infkEQˆ
Let Wt, t ≥ 0 be a standard k-dimensional Brownian motion with its natural filtra-tion (P-zero sets added) Ft, t ≥ 0. The dynamics of the price process of the ith stock Si=
Let us suppose that there are as many risky assets as sources of randomness, that is, k=d. Then it is trivial that there exists a uniquely determinedd-dimensional, deterministic processθ=n
defines the unique elementQ ∈ M. It is straightforward to check that ρT is lognormally distributed both underP and underQ. In particular,ρT,1/ρT ∈ W, so Assumptions 4.1, 4.2 and 4.3 hold.
Any contingent claim inL1(Q)is replicable by the martingale representation theorem. It is trivial to see that there must be some0<ˆt < T for which
has a non-degenerate joint normal distribution. It is easy to see that G∗:=
is anFT-measurable and non-degenerate Gaussian random variable which is independent of Pd
i=1
RT
0 θi(s)dWsi and hence of ρT. Lemma 6.15 provides a uniformU∗ independent ofρT, that is, satisfying Assumption 4.4. When d = 1, σ 6= 0, µ ∈ Rconstants then we get the Black-Scholes model, see e.g. [15].
Remark 4.19. In [80] a new method for constructingX∗ in Theorem 4.18 was found which applies to all complete markets (see Section 1.3) and it allows to construct X∗ which is a function ofρonly. Hence, in the case of complete markets, the existence requirement ofU∗
can be dropped in Assumption 4.4.
There are also examples of incomplete markets satisfying Assumption 4.4 where Theo-rem 4.18 applies. However, the class of such models is rather narrow. We refer to [74] and [80].
Extending results of the present chapter to larger classes of incomplete models is a chal-lenge. Some progress in this direction has been made in [77].
5 Illiquid markets
In financial practice, trading moves prices against the trader: buying faster increases ex-ecution prices, and selling faster decreases them. This aspect of liquidity, known as market depth [16] or price-impact, is widely documented empirically [40, 30], and has received in-creasing attention, see [62, 10, 2, 97, 84, 45]. These models depart from the literature on frictionless markets, where prices are the same for any amount traded.
The growing interest in price-impact has also highlighted a shortage of effective theoret-ical tools. In discrete time, several researchers have studied these fundamental questions, [5, 70, 38, 69], but extensions to continuous time have proved challenging. In this chapter we shall prove an existence theorem for optimal strategies in a very general continuous-time model under the assumption that trading costs are superlinear functions of the trading speed.
This assumption is consistent with empirical data, see [30].
Superlinear frictions in the sense of the present dissertation entail that execution prices become arbitrarily unfavorable as traded quantities per unit of time grow: buying or selling too fast becomes impossible. As a result, trading is feasible only at finite rates – the number of sharesϕtwill be assumed absolutely continuous. This feature sets apart superlinear frictions from frictionless markets, in which the number of shares is merely predictable, see Sections 1.3 and 5.1.
This chapter is based on [47].
5.1 Model
For a finite time horizon T > 0, consider a continuous-time filtered probability space (Ω,F,(Ft)t∈[0,T], P)where the filtration is right-continuous and F0 coincides with the fam-ily ofP-zero sets. O denotes the optional sigma-field on Ω×[0, T], that is, the sigma-field generated by the family of c `adl `ag adapted processes. The market includes a riskless and per-fectly liquid assetS0 withSt0≡1,t∈[0, T], anddrisky assets, described by c `adl `ag, adapted processes(Sti)1≤i≤dt∈[0,T]. Henceforth S denotes thed-dimensional process with components Si, 1≤i≤d. The components of a(d+ 1)-dimensional vectorxare denoted byx0, . . . , xd.
The next definition identifies those strategies for which the number of shares changes over time at some finite rate.
Definition 5.1. A feasible strategy is a processφin the class A:=
(
φ:φis anRd-valued,O-measurable process, Z T
0 |φu|du <∞a.s.
)
. (154)
In this definition, the processφrepresents thetrading rate, that is, the speed at which the number of shares in each asset changes over time, and the conditionRT
0 |φu|du <∞means thatabsolute turnover (the cumulative number of shares bought or sold) remains finite in finite time. Define, for eachφ∈ A,
ϕt:=
Z t 0
φudu, t∈[0, T],
the number of stock in the portfolio at timetin the respective assets (integration is meant componentwise).
The above definition significantly differs from the one of admissible strategies in friction-less markets in Section 1.3: this definition restricts the number of shares to be (absolutely) continuous, while usual admissible strategies have an arbitrarily irregular number of shares.
Note also that the definition of feasibility does not involve the asset price at all.
Assume S to be a semimartingale and recall Section 1.3. Note that ϕ above is a pre-dictable, locally bounded process, hence it isS-integrable.
In the absence of frictions the value of a self-financing portfolio at timeT is z0+
Z T 0
ϕtdSt,
wherez0represents the initial capital, see (6). Note that the investor holdsϕT units of stock at the terminal date which is worthϕTST. Hence the value of his/her cash (bank account) position at the terminal date is
z0+ Z T
0
ϕtdSt−ϕTST = z0− Z T
0
Stφtdt, (155)
where we performed, formally, an integration by parts and recalledϕ0 = 0as well. Notice, however, that the right-hand side of (155) makes sense for any c `adl `agS and not only for semimartingales. Indeed, by the c `adl `ag property the function St(ω), t ∈ [0, T] is bounded for almost every ω ∈ Ω, hence the integral in question is finite a.s. for each φ satisfying RT
0 |φt|dt <∞a.s.
Now we look at how (155) changes in the presence of illiquidity. For a given trading strategyφ, frictions reduce the cash position, by making purchases more expensive, and sales less profitable. We model this effect by introducing a function G, which summarizes the impact of frictions on the execution price at different trading rates:
Assumption 5.2. LetG: Ω×[0, T]×Rd→R+be aO ⊗ B(Rd)-measurable function, such that G(ω, t,·)is convex withG(ω, t, x)≥G(ω, t,0)for allω, t, x. Henceforth, setGt(x) :=G(ω, t, x), i.e. the dependence onωis omitted, andtis used as a subscript.
Taking price-impact into accout, for a given strategyφ ∈ Aand an initial asset position z∈Rd+1, the resulting positions at timet∈[0, T]in the risky and safe assets are defined as:
Xti=Xti(z, φ) :=zi+ Z t
0
φiudu 1≤i≤d, (156)
Xt0=Xt0(z, φ) :=z0− Z t
0
φuSudu− Z t
0
Gu(φu)du. (157)
The first equation merely says that the cumulative number of sharesXti in thei-th asset is given by the initial number of shares, plus subsequent flows. The second equation, compared to (155), contains a new term involving the frictionG, which summarizes the impact of trad-ing on execution prices. The conditionG(ω, t, x)≥G(ω, t,0)means that inactivity is always cheaper than any trading activity. Most models in the literature assumeG(ω, t,0) = 0, but the above definition allows forG(ω, t,0)>0, which is interpreted as a cost of participation in the market, such as the fees charged by exchanges to trading firms. The convexity ofx7→Gt(x) implies that trading twice as fast for half the time locally increases execution costs – speed is expensive. Indeed, letg(x) = G(ω, t, x), i.e. focus on a local effect. Then, by convexity, g(x)≤(1−1/k)g(0) + (1/k)g(kx)fork >1, and therefore(g(kx)−g(0))T /k≥(g(x)−g(0))T, which means that increasing trading speed by a factor ofkand reducing trading time by the same factor implies higher trading costs, excluding the participation cost captured byg(0).
Finally note that, in general,Xt0may take the value−∞for some (unwise) strategies.
With a single risky asset and withG(ω, t,0) = 0, the above specification is equivalent to assuming that a trading rate ofφt6= 0implies an instantaneous execution price equal to
S˜t=St+Gt(φt)/φt (158)
which is (by positivity of G) higher than St when buying, and lower when selling. Thus, G ≡ 0 boils down to a frictionless market, while proportional transaction costs correspond toGt(x) =εSt|x| with someε > 0. Yet, we focus on neither of these settings, which entail either zero or linear costs, but rather on superlinear frictions, defined as those that satisfy the following conditions. Note that we require a strong form of superlinearity here (i.e. the cost functional grows at least as a superlinear power of the traded volume).
Assumption 5.3. There isα >1and a c `adl `ag processH such that
t∈[0,T]inf Ht>0 a.s., (159)
Gt(x)≥Ht|x|α, for allt, x; a.s. (160) Z T
0
sup
|x|≤N
Gt(x)
!
dt <∞ a.s. for allN >0. (161) Condition (160) is the central superlinearity assumption. Condition (159) requires that frictions never disappear, and (161) says that they remain finite in finite time for uniformly bounded trading rates. In summary, these conditions characterize nontrivial, finite, super-linear frictions. Note that (160) implies thatS˜t in (158) becomes arbitrarily negative asφt
becomes negative enough, i.e. when selling too fast.
Remark 5.4. Although we can treat a generalS, the most important case is whereS has non-negative components, and therefore a positive number of units of risky positions has positive value. Otherwise, ifS can take negative values, a larger number of units does not imply a position with higher value, but only a larger exposure to default.
Assume in the rest of this remark thatS is non-negative and one-dimensional (for sim-plicity). Take φ ∈ A and consider the (optional) set A := {(ω, t) : φt(ω) < 0, St(ω) + G(ω, t, φt(ω))/φt(ω) ≥ 0}, which identifies the times at which execution prices are positive.
Clearly,XTi(z, φ′)≥XTi(z, φ),i= 1,2forφ′t(ω) :=φt(ω)1A. Hence one may always replace the set of strategiesAby
A+:={φ∈ A:St(ω) +G(ω, t, φt(ω))/φt(ω)≥0whenφt(ω)<0},
without losing any “good” investment. In other words, we may restrict ourselves to trading strategies with positive execution prices at all times, because any other strategy is dominated pointwise by a strategy that trades at the same rate when the execution price is positive, and otherwise does not trade. The classA+may be economically more appealing as it excludes the unintended consequence of (160) thatSt(ω) +G(ω, t, φt(ω))/φt(ω)→ −∞wheneverφt(ω)→
−∞.
The most common example in the literature is, with one risky asset, the friction Gt(x) := Λ|x|αfor someΛ>0, α >1
(see e.g. [38]). Another possibility is Gt(x) := ΛSt|x|α. In multiasset models the friction Gt(x) := xTΛxfor some symmetric, positive-definite,d×dmatrixΛ has been suggested in [45].
Remark 5.5. Our results remain valid assuming that (160) holds for |x| ≥ M only, with someM > 0. Such an extension requires only minor modifications of the proofs, and may accommodate models for which a low trading rate incurs, for instance, either zero or linear costs.